Scalar multiplication and parallel vectors Problem set B

Given \(\mathbf{p}=\begin{pmatrix}-4\\2\\3\end{pmatrix}\) and \(\mathbf{q}=\begin{pmatrix}1\\-5\\6\end{pmatrix}\), find:

1. (a) \(2\mathbf{p}+5\mathbf{q}\)
2. (b) \(3\mathbf{p}-4\mathbf{q}\)

Given \(\mathbf{x}=\begin{pmatrix}6\\-2\end{pmatrix}\), \(\mathbf{y}=\begin{pmatrix}2\\7\end{pmatrix}\) and \(\mathbf{z}=\begin{pmatrix}-4\\0\end{pmatrix}\), find:

1. (a) \(2\mathbf{x}+\mathbf{y}-2\mathbf{z}\)
2. (b) \(5\mathbf{x}-3\mathbf{y}+4\mathbf{z}\)

Given \(\mathbf{u}=\begin{pmatrix}3\\-4\\9\end{pmatrix}\) and \(\mathbf{v}=\begin{pmatrix}15\\p\\q\end{pmatrix}\), find the values of \(p\) and \(q\) so that \(\mathbf{v}\) is parallel to \(\mathbf{u}\).

Given \(\mathbf{r}=\begin{pmatrix}4\\-6\\10\end{pmatrix}\) and \(\mathbf{s}=\begin{pmatrix}p\\9\\q\end{pmatrix}\), find the values of \(p\) and \(q\) so that \(\mathbf{s}\) is parallel to \(\mathbf{r}\).

PQRS is a parallelogram with \(\overrightarrow{PQ}=\mathbf{u}\) and \(\overrightarrow{PS}=\mathbf{v}\). K is the midpoint of PQ. L is on the line SR extended beyond R, with RL = 2SR. Write these in terms of \(\mathbf{u}\) and \(\mathbf{v}\).

1. (a) \(\overrightarrow{PK}\)
2. (b) \(\overrightarrow{SL}\)
3. (c) \(\overrightarrow{KL}\)

PQRS is a parallelogram with \(\overrightarrow{PQ}=\mathbf{u}\) and \(\overrightarrow{PS}=\mathbf{v}\). X is on the line PS extended beyond S, with SX = \(\frac{1}{2}\)PS. Y is the midpoint of QR. Write these in terms of \(\mathbf{u}\) and \(\mathbf{v}\).

1. (a) \(\overrightarrow{QY}\)
2. (b) \(\overrightarrow{XR}\)
3. (c) \(\overrightarrow{XY}\)

Two forces on a crate are \(\mathbf{f}=\begin{pmatrix}3\\-4\\6\end{pmatrix}\) N and \(\mathbf{g}=\begin{pmatrix}2\\5\\-1\end{pmatrix}\) N.

1. (a) A third force is \(4\mathbf{f}-\mathbf{g}\). Find it as a column vector.
2. (b) A fourth force \(\mathbf{w}=\begin{pmatrix}9\\p\\q\end{pmatrix}\) N acts along the same line as the first force. Find \(p\) and \(q\).

The vectors \(\begin{pmatrix}k\\5\end{pmatrix}\) and \(\begin{pmatrix}20\\k\end{pmatrix}\) are parallel. Find the two possible values of \(k\).

Show that \(\mathbf{u}=\begin{pmatrix}-10\\4\\6\end{pmatrix}\) and \(\mathbf{v}=\begin{pmatrix}15\\-6\\-9\end{pmatrix}\) are parallel, and state whether they point the same way.